MCQ
$\int_0^\infty {{e^{ - 2x}}(\sin 2x + \cos 2x)\,dx = } $
- A$1$
- B$0$
- ✓$\frac{1}{2}$
- D$\infty $
$ = \left[ { - {e^{ - x}}\frac{{\cos 2x}}{2}} \right]_0^\infty - \int_0^\infty {\left( { - 2{e^{ - 2x}}} \right)\,} \left( {\frac{{ - \cos 2x}}{2}} \right){\rm{ }}dx$
$ + \int_0^\infty {{e^{ - 2x}}\cos 2x\,dx} $
$ = \frac{1}{2}$.
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Statement $-1 :$ $S=\{x:f(x)=f^{-1}(x)\}=\left\{ {0, - 1} \right\}$
Statement $-2 :$ $ f $ is a bijection.
$x+y+z=1$
$10 x+100 y+1000 z=0$
$q r x+p r y+p q z=0$.
| $List-I$ | $List-II$ |
| ($I$) If $\frac{q}{r}=10$, then the system of linear equations has | ($P$) $x=0, y=\frac{10}{9}, z=-\frac{1}{9}$ as a solution |
| ($II$) If $\frac{ p }{ r } \neq 100$, then the system of linear equations has | ($Q$) $x =\frac{10}{9}, y =-\frac{1}{9}, z =0$ as a solution |
| ($III$) If $\frac{p}{q} \neq 10$, then the system of linear equations has | ($R$) infinitely many solutions |
| ($IV$) If $\frac{p}{q}=10$, then the system of linear equations has | ($S$) no solution |
| ($T$) at least one solution |
The correct option is: